Question REF  corresponds in this restricted context to the following statement, proved in [1]} (for the existence part) and [2]} (for the uniqueness): given a closed surface \(S\)  of genus at least 2 and two smooth Riemannian metrics \(h_-, h_+\)  of curvature \(K>-1\)  on \(S\) , there exists a unique quasifuchsian manifold \(M\)  homeomorphic to \(S\times {\mathbb {R}}\)  and a unique geodesically convex subset \(\Omega \subset M\)  such that the induced metrics on the two boundary components of \(\Omega \)  are isotopic to \(h_-\)  and \(h_+\) . Similarly, [2]} contains an answer to Question REF  in the quasifuchsian setting: given two smooth metrics \(h_-, h_+\)  on \(S\)  of curvature \(K<1\)  with closed, contractible geodesics of length \(L>2\pi \) , there exists a unique quasifuchsian manifold and a unique geodesically convex subset \(\Omega \subset M\)  such that the third fundamental forms on the two boundary components of \(\Omega \)  are isotopic to \(h_-\)  and \(h_+\) .
